Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1608.03089 · Scientific Reports

Multi-observable Uncertainty Relations in Product Form of Variances

Hui-Hui Qin, Shao-Ming Fei, Xianqing Li-Jost

Published Aug 8, 2016• 37 citations
arXivHTMLPDF

Abstract

We investigate the product form uncertainty relations of variances for n (n ≥ 3) quantum observables. In particular, tight uncertainty relations satisfied by three observables has been derived, which is shown to be better than the ones derived from the strengthened Heisenberg and the generalized Schrödinger uncertainty relations, and some existing uncertainty relation for three spin-half operators. Uncertainty relation of arbitrary number of observables is also derived. As an example, the uncertainty relation satisfied by the eight Gell-Mann matrices is presented.

DOI
10.1038/srep31192

Related papers

Variance uncertainty relations without covariances for three and four observables

V. V. Dodonov

Sum and product uncertainty relations, containing variances of three or four observables, but not containing explicitly their covariances, are derived. Their consequences are, in particular, inequalities, giving nonzero lower bounds for the products of two variances in the case of zero mean value of the commutator between the related operators. Explicit examples show that the bounds can be better than the known Robertson-Schr\"odinger one.

arXiv:1711.04037•Physical Review A•Feb 7, 2018•30 citations

A Note on the Strong Unitary Uncertainty Relations

Ting-Gui Zhang

Uncertainty relations satisfied by the product of variances of arbitrary n oservables have attracted much attention.In a recent article [Phys.Rev.Lett.120,230402(2018)], the authors provided so called strong unitary uncertainty relations for a set of unitary matrices by using the positivity property of the Gram matrix, from which uncertainty relations satisfied by two quantum mechanical observables are obtained.We derive the explicit uncertainty relations satisfied by n quantum mechanical observables from such Gram matrix approach.By some algebraic transformations, we show that these uncertainty relations are just the same as the ones derived from a positive semi-definite Hermitian matrix generated by the mean values of n observables in [Scientific Reports 6,31192 (2016)].

Journal of Applied and Theoretical Physics Research•Feb 20, 2019•0 citations

Experimental investigation of a stronger multi-observable uncertainty relation with single photons

Kunkun Wang, Peng Xue, Bowen Fan +1 more

The Heisenberg-Robertson uncertainty relation presents a lower bound on the product of variance of two observables, and provides a trade-off relation of measurement errors of these two observables for any given quantum states. Recently multi-observable uncertainty relations relating on that product of variances are proposed. Here we experimentally demonstrate these multi-observable uncertainty relations are valid in a state-dependent manner and the lower bound is tight for multi-observable being incompatible on the state of the system being measured. We find the behaviour of multiple incompatible observables agrees with the predictions of quantum theory.

Quantum Communications and Quantum Imaging•Sep 18, 2018•3 citations

Uncertainty relations for several observables via the Clifford algebras

V. V. Dodonov

New sum and product uncertainty relations, containing variances of up to five observables, but not containing explicitly their covariances, are derived. New inequalities for three observables, especially for the angular momentum and spin-1/2 operators, are also presented.

Journal of Physics Conference Series•Apr 1, 2019•7 citations

A Stronger Multi-observable Uncertainty Relation

Qiu-Cheng Song, Jun-Li Li, Guang-Xiong Peng +1 more

Uncertainty relation lies at the heart of quantum mechanics, characterizing the incompatibility of non-commuting observables in the preparation of quantum states. An important question is how to improve the lower bound of uncertainty relation. Here we present a variance-based sum uncertainty relation for N incompatible observables stronger than the simple generalization of an existing uncertainty relation for two observables. Further comparisons of our uncertainty relation with other related ones for spin- and spin-1 particles indicate that the obtained uncertainty relation gives a better lower bound.

arXiv:1701.01072•Scientific Reports•Mar 20, 2017•47 citations

Tighter uncertainty and reverse uncertainty relations

Debasis Mondal, Shrobona Bagchi, Arun Kumar Pati

We prove a few state-dependent uncertainty relations for the product as well as the sum of variances of two incompatible observables. These uncertainty relations are shown to be tighter than the Robertson-Schr\"odinger uncertainty relation and other ones existing in the current literature. Also, we derive a state-dependent upper bound to the sum and the product of variances using the reverse Cauchy-Schwarz inequality and the Dunkl-Williams inequality. Our results suggest that not only can we not prepare quantum states for which two incompatible observables can have sharp values, but also we have both the lower and the upper limits on the variances of quantum mechanical observables at a fundamental level.

arXiv:1607.06712•Physical Review A•May 18, 2017•54 citations
We investigate the product form uncertainty relations of variances for n (n ≥ 3) quantum observables. In particular, tight uncertainty relations satisfied by three observables has been derived, which is shown to be better than the ones derived from the strengthened Heisenberg and the generalized Schrödinger uncertainty relations, and some existing uncertainty relation for three spin-half operators. Uncertainty relation of arbitrary number of observables is also derived. As an example, the uncertainty relation satisfied by the eight Gell-Mann matrices is presented.
Uncertainty relations satisfied by the product of variances of arbitrary n oservables have attracted much attention.In a recent article [Phys.Rev.Lett.120,230402(2018)], the authors provided so called strong unitary uncertainty relations for a set of unitary matrices by using the positivity property of the Gram matrix, from which uncertainty relations satisfied by two quantum mechanical observables are obtained.We derive the explicit uncertainty relations satisfied by n quantum mechanical observables from such Gram matrix approach.By some algebraic transformations, we show that these uncertainty relations are just the same as the ones derived from a positive semi-definite Hermitian matrix generated by the mean values of n observables in [Scientific Reports 6,31192 (2016)].
The Heisenberg-Robertson uncertainty relation presents a lower bound on the product of variance of two observables, and provides a trade-off relation of measurement errors of these two observables for any given quantum states. Recently multi-observable uncertainty relations relating on that product of variances are proposed. Here we experimentally demonstrate these multi-observable uncertainty relations are valid in a state-dependent manner and the lower bound is tight for multi-observable being incompatible on the state of the system being measured. We find the behaviour of multiple incompatible observables agrees with the predictions of quantum theory.
Uncertainty relation lies at the heart of quantum mechanics, characterizing the incompatibility of non-commuting observables in the preparation of quantum states. An important question is how to improve the lower bound of uncertainty relation. Here we present a variance-based sum uncertainty relation for N incompatible observables stronger than the simple generalization of an existing uncertainty relation for two observables. Further comparisons of our uncertainty relation with other related ones for spin- and spin-1 particles indicate that the obtained uncertainty relation gives a better lower bound.
We prove a few state-dependent uncertainty relations for the product as well as the sum of variances of two incompatible observables. These uncertainty relations are shown to be tighter than the Robertson-Schr\"odinger uncertainty relation and other ones existing in the current literature. Also, we derive a state-dependent upper bound to the sum and the product of variances using the reverse Cauchy-Schwarz inequality and the Dunkl-Williams inequality. Our results suggest that not only can we not prepare quantum states for which two incompatible observables can have sharp values, but also we have both the lower and the upper limits on the variances of quantum mechanical observables at a fundamental level.